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DISCUSSION PAPER SERIES

ABCD

www.cepr.org

Available online at:

www.cepr.org/pubs/dps/DP6417.asp

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No. 6417

ON THE OPTIMAL NUMBER

OF REPRESENTATIVES

Emmanuelle Auriol and Robert J. Gary-Bobo

DEVELOPMENT ECONOMICS and

PUBLIC POLICY

ISSN 0265-8003

ON THE OPTIMAL NUMBER

OF REPRESENTATIVES

Emmanuelle Auriol, IDEI, Toulouse School of Economics and ARQADE

Robert J. Gary-Bobo, Université Paris 1 Panthéon-Sorbonne, Paris School of

Economics and CEPR

Discussion Paper No. 6417

August 2007

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Copyright: Emmanuelle Auriol and Robert J. Gary-Bobo

CEPR Discussion Paper No. 6417

August 2007

ABSTRACT

On the Optimal Number of Representatives*

We propose a normative theory of the number of representatives based on a

stylized model of a representative democracy. We derive a simple formula, a

"square-root theory" which gives the number of representatives in parliament

as proportional to the square root of total population. Simple econometric tests

of the formula on a sample of a 100 countries yield surprisingly good results.

These results provide a benchmark for a discussion of the appropriateness of

the number of representatives in some countries. It seems that the United

States have too few representatives, while France and Italy have too many.

The excess number of representatives matters: it is positively correlated with

indicators of red tape, barriers to entrepreneurship and perceived corruption.

JEL Classification: D7, H11 and H40

Keywords: constitution design, incentives, number of representatives and

representative democracy

Emmanuelle Auriol

IDEI

University of Toulouse I

32 rue de Mounic

09340 Verniolle

FRANCE

Email: eauriol@cict.fr

For further Discussion Papers by this author see:

www.cepr.org/pubs/new-dps/dplist.asp?authorid=127618

Robert J. Gary-Bobo

Université Paris 1

Panthéon-Sorbonne

Maison des Sciences Economiques

106-112 Boulevard de l'Hôpital

75647 Paris Cedex 13

FRANCE

Email: garybobo@univ-paris1.fr

For further Discussion Papers by this author see:

www.cepr.org/pubs/new-dps/dplist.asp?authorid=125284

* We thank Nabil Al Najjar, David Austen-Smith, Tim Besley, Helge Berger,

Jacques Drèze, Gabrielle Demange, Jean-Pierre Florens, Françoise Forges,

Roger Guesnerie, Arye Hillman, Mamoru Kaneko, Jean-Jacques Laffont,

Jean-François Laslier, John Ledyard, Philippe Mongin and Thomas Palfrey for

their help and comments, as well as seminar audiences at the Free University

of Berlin, UC Dublin, Universities of Tilburg, Toulouse, Stockholm, University

of Wisconsin, Madison, University of Illinois at Urbana-Champaign,

Northwestern University, Boston College, Caltech, UCLA, and CEPR (Public

Policy Program) for useful remarks. The present paper is a deeply revised and

extended version of a manuscript circulated under the same title in the year

2000 (DP no 1286 of the Center for Mathematical Studies, in Economics and

Management Sciences, Northwestern University).

Submitted 25 July 2007

1 Introduction

The production of public goods aﬀects the well-being of large number of citizens, whereas a

typically much smaller number of individuals is in charge of public decisions. This is true at

almost all levels of society: there are parliaments at the national level, councils at the local

levels and even committees within public or private organizations. The presence of costs

associated with the acquisition of information and with the preparation of decisions plays

a major role in this concentration of power. The forces driving the division of labor help

understanding the emergence of representatives. As a counterpart, protection against the

opportunistic behavior of these representatives becomes a major justiﬁcation of collective de-

cision rules. This paper studies the trade-oﬀ between the need to economize on decision costs,

suggesting that a small number of individuals should specialize in public decision-making,

and the democratic requirement that decisions should reﬂect the citizens’ true preferences.

We focus on a theory of the optimal number of representatives, that we also test with the

help of political data.

We adopt a two-stage approach to constitutional design,

1

with a constitutional and

a legislative stage, to derive the optimal number of representatives. In contrast to most of

the recent work on constitution design, we completely black-box elections and voting and

construct what could be called a reduced-form theory of representative democracy. The leg-

islators’ assembly is modeled as a random sample of preferences, drawn from the population

of citizens. The randomly chosen representatives do not vote; they use a nonmanipulable,

revealing mechanism instead. This mechanism reveals the representatives’ preferences and

eﬃcient public-decisions are carried out by a self-interested executive. During the prelimi-

nary constitutional stage, ﬁctitious Founding Fathers choose decision rules behind the veil

of ignorance, so as to maximize the expected total sum of citizens’ utility. The Founding

Fathers know that no agent is benevolent. It follows from this that the executive’s hands

must be tied as much as possible and that representatives must be provided with incentives

to reveal preferences truthfully. In addition, our Founding Fathers know that they don’t

1

On this question, see the survey in Mueller (2003), and the discussion of some recent contributions below.

2

know the distribution of preferences that will prevail in society. The novelty of this article

is that we do not assume that this distribution is common knowledge. A robust mechanism

is therefore required, in the following particular sense: among nonmanipulable mechanisms,

the Founding Fathers pick a decision rule that maximizes expected utility against a vague (or

noninformative) prior relative to citizens’ preferences. Using a well-known technique from

Bayesian statistics, a limiting argument is used to derive the eﬀect of the Founding Fathers’

ignorance on the optimal mechanism.

2

Robustness in this sense can be understood as a political stability requirement. The

Founding Fathers know that society is going to evolve, but they cannot anticipate in which

way. A constitution could not last for more than 200 years if it was tailored too closely to a

particular preference proﬁle. It must work under very diﬀerent distributions of preferences.

Our model singles out a well-deﬁned robust mechanism, that happens to be a sampling

Groves mechanism. Statistical sampling properties then yield an optimal sample size, which

trades oﬀ the direct and opportunity costs of representatives with the welfare loss induced

by representation (i.e., the loss due to the fact that a subset of citizens make decisions).

A “square-root formula” for the optimal number of representatives directly follows

from this stylized model of representation. The rule is then tested with the help of a sample

of more than 100 countries, and we ﬁnd that our square-root theory is almost true and

reasonably robust. World data is well-approximated by a number of national representatives

proportional to N

0.4

, where N is the country’s total population. We also identify the US,

France and Italy as outliers. The former lie below the regression line; the latter two much

above. The same model does not ﬁt the data on the 50 US State Legislatures very well. We

conclude that the historical rigidity of political US institutions is likely to be the cause of this

lower quality of ﬁt. Indeed constitutional History shows that the representation ratio has

constantly decreased for more than 200 years in the United States. Tocqueville (1835, part I,

Chap. VIII, p 190, footnote) already noted the fact that the representation ratio decreased

from 1 representative for every 30,000 inhabitants in 1792, to 1 over 48,000 in 1832. This

trend has not been reversed ever since, the ratio reaching a record low of 1 over 611,000

2

The most technical aspects of our theory are presented in Auriol and Gary-Bobo (2007).

3

in the recent years. Furthermore, the number of seats in the House of Representatives has

reached a ceiling of 435 in 1910.

3

According to our results, the US Lower and Upper Houses

should have a total of 807 members.

We ﬁnally check for correlation of the number of representatives with some indices

measuring openness to trade, the costs of setting up a new ﬁrm (i.e. “red tape”), the degree

of state interference in markets, and perceived corruption.

4

The results are clearly that the

number of representatives matters: it is positively and signiﬁcantly correlated with state

interference, red tape, and corruption. More precisely, we cannot reject the fact that it is the

excess number of representatives (i.e., the actual number less the number predicted by the

N

0.4

formula) which in fact matters for red tape and the degree of state interference.

The question of the appropriate number of seats in US Parliament has been posed a

long time ago by the founding fathers and opponents of the American Constitution. James

Madison addressed the question in a famous passage of Federalist n

◦

10:

In the ﬁrst place, it is to be remarked that however small the Republic may be,

the Representatives must be raised to a certain number, in order to guard against

the cabals of a few; and however large it may be, they must be divided to certain

number, in order to guard against the confusion of a multitude.

Madison, Federalist 10 (in Pole (1987), p 155).

The Anti-Federalist writers have emphasized a related point:

The very term, representative, implies, that the person or body chosen for this

purpose, should resemble those who appoint them (...). Those who are placed in-

stead of the people, should possess their sentiments and feelings, and be governed

by their interests, or, in other words, should bear the strongest resemblance of

those in whose room they are substituted. (...) Sixty-ﬁve men cannot be found in

3

This number has been ﬁxed by statute in 1929. See O’Connor ans Sabato (1993), p 191.

4

We use indices constructed by Barro and Lee (1994), Djankov et al. (2002), Treisman (2000), and

Transparency International, respectively.

4

the Unites States, who hold the sentiments, possess the feelings, or are acquainted

with the wants and interests of this vast country.

Essays of Brutus, III, 1787 (in Storing (1981), p 123)

Some essential ideas are condensed in the above quotations. In the paper we interpret

them as stating that there exists a tradeoﬀ between the need to protect citizens against the

dictatorship of a minority and that of reducing the costs of public decision-making. As

far as we know the problem of the optimal number of legislators has been studied by a

handful of economists only.

5

In contemporary writings, Buchanan and Tullock (1962) are

clearly the forerunners of the approach followed here. Thinking about constitutional design,

they developed a theory of the optimal constitution based on 4 variables: rules for choosing

representatives; rules for deciding issues in assemblies; the degree of representation (i.e., the

proportion of total population elected); and the basis of representation (i.e., for instance,

the geographical basis). Buchanan and Tullock’s approach is clearly normative, insofar as

the goal of the analysis is to ﬁx the 4 variables in order to minimize the expected sum of

decision-making and external costs of institutions. Another forerunner is Stigler (1976), who

sketched a theory of the degree of representation and proposed some regression work on the

number of representatives in relation to total population in the US States.

A small (but inﬂuential) number of authors belonging to the Public Choice school

has played with the ideas emphasized here a long time ago: following Dahl (1970), Mueller

et al. (1972) discuss random representation. Tullock (1977) went as far as to ponder over

the practical possibility of using pivotal mechanisms in the US Congress to make public

decisions. In the present paper, our intention is not to advocate the recourse to random choice

of legislators, or Groves mechanisms in practice, but to propose a model of representative

democracy in reduced form and to derive a formula for the optimal number of representatives.

We are not the ﬁrst to adopt a “reduced-form approach” to models politics. For

instance, in Becker (1983), political parties and voting receive little attention because “they

5

This problem is essentially distinct from that of fair representation or apportionment, that has been

studied quite extensively, e.g. Balinski and Young (2001). Our theory is not related to L. S. Penrose’s

(1946) square-root formula. Penrose’s formula determines the size of a country’s delegation in supra-national

institutions like the UN or EU, not the number of representatives itself.

5

are assumed mainly to transmit the pressure of active groups”. Becker (1983) deﬁnes political

equilibrium as a Nash equilibrium among pressure groups using expenditures on inﬂuence

as strategic variables. Such an approach has both limitations and advantages. More recent

contributions in which a common agency model is used to study public policy-making can

also be viewed as employing a reduced-form methodology (see e.g. Dixit et al. (1997)).

There has been a recent revival of interest in the normative method among writers

in Political Economy, Voting Theory and Mechanism Design. Our normative approach does

not rely on the existence of a benevolent planner and our self-interested executives are

clearly in line with the citizens-candidate approach of Osborne and Slivinski (1996) and

Besley and Coate (1998). The two-stage approach to Constitutional Design recently received

further impetus from Aghion and Bolton (2003), Barbera and Jackson (2004), Erlenmaier

and Gersbach (2001). Some contributions explore voting rules, or alternative collective

decision procedures, with the idea of improving eﬃciency through a better expression of the

intensity of preferences (e.g., Casella 2005). An extension of optimal taxation theory to a

dynamic setting in which citizens must rely on a non-benevolent politician to implement

redistribution policies has been proposed by Acemoglu et al. (2007). Political Economy

considerations are now more and more introduced in normative analysis and give rise to new

constraints, in addition to revelation or incentive constraints. Our underlying philosophy

has much in common with that of these recent contributions.

In the following, Section 2 presents our basic assumptions; Section 3 develops our

model of representation; Section 4 derives the robust representation mechanism and the

square-root theory of the optimal number of representatives. Section 5 presents the empirical

results: econometric tests of the square-root theory in the world and among the US State

legislatures; it also discusses the empirical relevance of the number of representatives by

showing its impact on red tape, state interference and corruption indices. A few technical

results are proved in the appendix.

6

2 The Model

2.1 Basic Assumptions

We consider an economy composed of N + 1 agents, indexed by i = 0, 1, ..., N. A public

decision, denoted q, must be chosen in a set Q. Agent i will pay a tax denoted t

i

. This tax

must be interpreted as a subsidy if it is negative. Each agent’s utility depends on the public

decision and the tax.

Assumption 1. (Quasi-linearity) Utilities are quasi-linear, and deﬁned as v

i

(q) − t

i

, where

v

i

, is a private valuation function.

These valuation functions can be viewed as random draws in some unknown probability

distribution P on a set of admissible valuation functions V .

Assumption 2. (Statistical Independence) For all i, the v

i

are independent drawings from

the same distribution P on V . The distribution P has a well-deﬁned mean.

Society comprises three types of individuals. Agent i = 0, called the executive, is in charge of

executing the collective decision q. After some relabelling if necessary, agents i = 1, ..., n are

representatives; and agents i = n + 1, ..., N are passive citizens. The task of representatives

is to transmit information on preferences. We assume the following.

Assumption 3. (Cost of Representation) Each representative pays a ﬁxed cost F , i.e., if i

is a representative, then i’s utility is v

i

(q) − t

i

− F .

This cost can be viewed as the sum of direct and opportunity costs of becoming a represen-

tative or, alternatively, as an elementary form of information-acquisition cost paid by agent

i to obtain information about one’s own preferences v

i

. Under the former interpretation,

citizens use resources to transmit information to the collective decision system. Under the

latter interpretation, individuals do not know their own utility function and must incur costs

to become aware of their own preferences. The two interpretations are compatible.

A representation is basically a random sample of n ≤ N agents (or, equivalently, a

random sample of preferences v = (v

1

, ..., v

n

)).

7

Assumption 4. (Perfect Representation) The n representatives are independent random

drawings in the probability distribution P .

In practice, it is doubtful that voting mechanisms would produce an unbiased random sample

of preferences. On the one hand, Assumption 4 might seem a rather naive idealization, but

can be defended if our goal is to construct a normative theory of representative democracy

and to determine the optimal number of representatives. On the other hand, the idea

of unbiased random representation provides a desirable simpliﬁcation, putting the entire

electoral process in a black box. Representation by lot existed in some societies of the

past (see Hansen (1991), Manin (1997));

6

it has been discussed by political scientists (Dahl

(1990)) and is still used to select juries in some countries. However, the representation biases

induced by voting systems cannot be studied with the simplest form of this model. We will

nevertheless continue to work with this convenient idealization.

Deﬁnition 1 (Representation Mechanism). A representation mechanism is an array of

functions (f, t), where f is a collective decision rule mapping representatives’ reports about

preferences bv = (bv

1

, ..., bv

n

) into Q, i.e., q = f(bv), and a list of tax functions denoted t =

(t

0

, t

1

,...t

N

), satisfying the budget constraint

P

N

i=0

t

i

= 0.

By deﬁnition, the constitution speciﬁes (f, t) for every possible value of n, but n itself is not

ﬁxed in the constitution.

2.2 The First-Best Optimum

We can now compute the ﬁrst-best optimum in the above deﬁned economy. The standard

Utilitarian, ﬁrst-best Bayesian decision maximizes the function

EW = E

P

(

N

X

i=0

(v

i

(q) − t

i

) | (bv

1

, ..., bv

n

)

)

− nF, (1)

6

The ancient greeks, in Athens, used random drawings to choose their legislators. The Athenian people’s

assembly itself, with its 6000 members, was in fact a random sample of the citizen population. Each citizen

attending a session of this Assembly would receive the equivalent of a worker’s daily wage. Socrates was

sentenced to death by a jury of 501 randomly drawn citizens (see Hansen (1991)).

8

with respect to q in Q, subject to the budget constraint

P

N

i=0

t

i

= 0, where E

P

denotes the

expectation with respect to probability P . Given that individual preferences are independent

draws in probability distribution P , this is equivalent to solving the problem:

max

q∈Q

(

(N + 1 − n)E

P

(v(q)) +

n

X

i=1

bv

i

(q) − nF

)

, (2)

where E

P

(v(.)) is the average utility function in the population. To understand how this

ﬁrst-best optimum looks like, assume for example that preferences are quadratic, with a

single-dimensional parameter θ, i.e., v

i

(q) = θ

i

q − q

2

/2 and that q is a nonnegative real

number. Assume in addition that P is such that E(θ) = µ and V ar(θ) = σ

2

. With these

speciﬁcations, (2) becomes

max

q∈Q

(

q

"

n

X

i=1

b

θ

i

+ (N + 1 − n)µ

#

− (N + 1)

q

2

2

− nF

)

. (3)

This immediately yields the optimal decision

q

∗

= f

∗

(

b

θ

1

, ...,

b

θ

n

) =

1

N + 1

n

X

i=1

b

θ

i

+ (N + 1 − n)µ

!

; (4)

Substituting (4) into EW , taking the expectation with respect to the distribution of θ

i

,

yields the ex ante expected welfare associated with the optimal decision rule f

∗

. After some

easy computations, we obtain

EW (f

∗

) =

nσ

2

2(N + 1)

+

(N + 1)µ

2

2

− nF, (5)

where we make use of the fact that the

b

θ

i

are i.i.d. This function being linear with respect

to n, we can state the following result.

Proposition 1. With quadratic preferences, the ﬁrst-best optimum has two possible values:

either n

∗

= N + 1, if σ

2

> 2(N + 1)F , (i.e., a Direct Democracy), or n

∗

= 0, if σ

2

≤

2(N + 1)F , (i.e., a ”Reign of Tradition”).

The interpretation of Proposition 1 is easy. If the dispersion of preferences is large

enough with respect to costs of representation, then direct democracy is ﬁrst-best optimal.

9

In other words, if the individual cost of participating in the collective decision process F

is small, or if the number of citizens is small, then democracy must be direct. The only

other case is not a democratic constitution: we call this “Reign of Tradition” because it

is not dictatorship (which would correspond to n = 1). In the Reign of Tradition, no

citizen is endowed with the power of deciding on behalf of others and we can view the

public decision as being the result of “Tradition”, i.e., f

∗

= µ. Another equivalent view

is that the decision is made by a disembodied benevolent planner. This arrangement is

optimal only if the dispersion of preferences is small or if the population is large and if in

addition, the prior mean of preference parameters µ is common knowledge. Proposition 1 is

disappointing, because it never prescribes a representative democracy, in which the solution

would be interior, i.e., 0 < n

∗

< N + 1. The most likely case is one in which F is small

but nonnegligible, N is very large, and tastes do not diﬀer in an extreme way, which seems

to indicate that the Reign of Tradition would always be the recommended solution. This

failure to pick a representative democracy as a solution is not essentially due to the fact

that expected welfare is linear with respect to n under quadratic preferences (and to the

fact that total representation costs nF are linear). It stems from the assumption that the

distribution of preferences is common knowledge. Indeed, if this is the case, if in addition

N is large and if the dispersion of tastes is reasonable, by the Law of Large Numbers, µ

is an excellent estimator of the true population-mean of individual valuations and it is not

useful to ask citizens about their taste parameters. Our claim is that there is something

wrong with the above deﬁnition of the optimum, because the model describes a world in

which information is not really decentralized. The model is that of an abstract benevolent

planner, endowed with prior knowledge of the distribution of preferences (i.e., (µ, σ) in the

quadratic example), but in a large economy with quadratic preferences, knowing µ means

knowing almost everything that is useful: Democracy is useless.

In Section 4 below, we propose a diﬀerent model in which information is fully decen-

tralized, the distribution of tastes is not common knowledge and democratic representation

is a useful (and only) way of producing information. Section 3 will ﬁrst provide some basic

deﬁnitions and pose the representatives’ incentive compatibility problem.

10

3 Representation and Incentives

To give formal content to the idea of an impartial and benevolent point of view on society,

we assume the existence of ﬁctitious agents called the Founding Fathers (hereafter the FF).

The FF are in charge of writing the constitution; they are assumed benevolent, Bayesian,

and Utilitarian, and they do nothing in the economy, apart from setting constitutional rules.

These FF know that, once the set of rules embodied in the constitution will be applied,

there will not exist a single omniscient, impartial and benevolent individual to carry out

public decisions. A disembodied “social planner” is not assumed to play an active role. This

imposes restrictions on the set of admissible mechanisms, described in sub-section 3.1. The

ensuing preference revelation problem is studied in sub-section 3.2.

3.1 Basic Constitutional Principles

The FF apply some important principles. First, Separation of Power holds: the executive

cannot be a representative. Second, a Subsidiarity Principle applies. According to Deﬁnition

1 above, a representation mechanism is an array of functions (f, t). To work in practice, such

a mechanism needs to be fully speciﬁed and this speciﬁcation may depend on a number of

controls or parameters. We need to allocate the power to choose the exact value of these

parameters, and these choices may open some possibilities of manipulation. This motivates

the following deﬁnition.

Deﬁnition 2 (Subsidiarity Principle). With the exception of the number of representatives

n itself, if the parameters needed to fully pin down and implement mechanism (f, t) are not

speciﬁed in the constitution and are not provided for by the representatives according to

constitutional rules, then they are chosen by the executive.

The Subsidiarity Principle simply says that the executive will ﬁll all the gaps in the public

decision process. It can of course be dangerous to let the executive choose crucial parameters

freely, because this executive is endowed with unknown preferences (v

0

is a random draw in

P ) and would be tempted to pursue private goals.

11

Third, the FF also apply a principle of “Anonymity” (or “Equality” in a weak sense),

which imposes equal treatment of indistinguishable individuals. This forces equal tax treat-

ment of all passive citizens, because their preferences are unknown (and there is no basis for

discrimination among them). Let t

0

denote the tax of agents i = n + 1, ..., N and i = 0. The

budget constraint can thus be rewritten as follows:

n

X

i=1

t

i

+ (N + 1 − n)t

0

= 0. (6)

3.2 Incentive Compatibility

The decision rule f, as well as taxes t, should be immune to manipulations of the rep-

resentatives and of the executive. Appealing to the Revelation Principle, we require the

representation mechanism (f, t) to be direct and revealing. But the agents’ beliefs about

others’ preferences are not common knowledge and are unknown to the FF. Mechanism

(f, t) must therefore be revealing whatever the beliefs of the representatives. In this con-

text, it almost immediately follows that (f, t) must be revealing in dominant strategies (see

Ledyard (1978)), i.e., for all i = 1, ...n, for all v

i

, bv

i

, and v

−i

,we must have

v

i

(f(v)) − t

i

(v) ≥ v

i

(f(bv

i

, v

−i

)) − t

i

(bv

i

, v

−i

),

where, as usual, we denote v

−i

= (v

1

, ..., v

i−1

, v

i+1

, ..., v

n

) and v = (v

i

, v

−i

).

Because of the subsidiarity principle, the self-interested executive could choose the free

parameters of (f, t) to maximise his (her) own utility v

0

. These parameters must therefore

be ﬁxed in the constitution. In our simple model, revelation in dominant strategies plus

”mast-tying” of the executive, put together, deﬁne non-manipulability.

Deﬁnition 3 (Non-Manipulability). A representation mechanism (f, t) is nonmanipulable if

it is revealing in dominant strategies and if all its parameters are speciﬁed in the constitution.

This deﬁnition means that, in addition to the revelation property, there are no free pa-

rameters that the executive could manipulate. It is possible to prove (see the appendix,

for comments and a formal statement), that under separation-of-powers, subsidiarity and

12

anonymity principles, non-manipulable mechanisms must assume the following form: the

decision rule f(.) must maximize an objective which is the sum of an arbitrary function k

and of the utilities reported by representatives, i.e.,

f(bv) ∈ arg max

q∈Q

(

k(q) +

n

X

i=1

bv

i

(q)

)

. (7)

And for all i = 1, ..., n, representatives must be subjected to the following tranfer schedules:

t

i

(bv) = −

X

j6=i

bv

i

(f(bv)) − k(f(bv)) + m(bv

−i

), (8)

where m is an arbitrary ﬁxed function that does not depend on bv

i

. Finally, arbitrary functions

k, and m must be ﬁxed in the constitution. Obviously, the choice of these crucial parameters

cannot be left to the executive, because the choice of k can distort decisions radically, while

the choice of m can distort transfers. We assume that the FF are constrained to choose

f(.) in this set of nonmanipulable mechanisms. When k ≡ 0, the class of nonmanipulable

mechanisms boils down to the well-known class of Clarke-Groves mechanisms, but restricted

to a random subset of agents called the representatives.

7

Note that these mechanisms are budget-balanced by construction, because there is

at least one citizen which is not a representative (i.e., at least agent 0 does not report

about his (her) preferences). In other words, passive citizens form a sink used to ﬁnance the

revelation incentives of the representatives. It follows that there are no ineﬃciencies due to

budget imbalance (budget surplus), as in the usual theory of pivotal mechanisms. The only

welfare losses are due to the fact that the information on preferences used by a representation

mechanism is not exhaustive; in other words, social costs are caused by sampling errors.

4 Robust Representation Mechanisms under Decen-

tralized Knowledge

The novelty of our approach is that we have assumed that the FF do not know the prob-

ability distribution of citizens’ preferences P , and they know that nobody knows it. We

7

On Groves mechanisms, see Clarke (1971), Groves (1973), Green and Laﬀont (1979), Holmstrom (1979),

Moulin (1986). On sampling Groves mechanisms, see Green and Laﬀont (1977), Gary-Bobo and Jaaidane

(2000).

13

add the constraint of decentralized knowledge to the assumptions of asymmetric information

and individual opportunism: the probability distribution of preferences P is not common

knowledge.

The fact that the FF do not know P poses a problem because they cannot fully specify

the expected (or average) welfare function that they would like to maximize by means of

the choice of a constitution. There are several ways of modeling behavior under ignorance

in decision theory. One is to use a non-probabilistic representation and a maximin principle

or, some more sophisticated variant in which the decison-maker uses a set of probability

distributions. The constitution would then be chosen so as to maximize welfare against the

worst-case scenario. Another approach is to choose decision rules that are optimal against a

non-informative, or vague prior. In contrast, this is a purely Bayesian approach. We choose

this latter route here. There is a mathematical diﬃculty in the representation of a decision

maker’s complete prior ignorance because a uniform distribution on the real line (or on the

set of integers) doesn’t exist.

8

It follows that a situation of complete prior ignorance can be

approached by limiting arguments, letting the prior’s variance go to inﬁnity.

4.1 The Founding Fathers’ Objective

We ﬁrst assume that the FF constrain themselves to choose a decision rule that satisﬁes

“Weak Utilitarianism”.

9

Deﬁnition 4 (Weak Utilitarianism). The decision rule f should maximize the expected

utility E

P

0

(v(q)) with respect to q in Q for some probability distribution P

0

on V .

Imposing Weak Utilitarianism in the sense of Deﬁnition 4 means that the decision rule must

maximize some weighted sum of utilities. Given that the FF are assumed to be Utilitarians,

this requirement is very weak, because P

0

can be chosen arbitrarily.

We can now derive what we call robust mechanisms. It is easy to see that, under

non-manipulability, the FF’s goal is essentially to choose the arbitrary function k. If the set

8

Bayesian statisticians have developed the theory of improper priors. See e.g. Bernardo and Smith (1994).

9

But the utilitarian principle could also be derived, in the manner of Harsanyi (1955), by assuming that

the FF are rational decision-makers, and choose the objective function behind the veil of ignorance.

14

of possible utilities V is convex, the weak utilitarianism requirement imposes to choose k of

the form k(q) = bv

0

(q), where b ≥ 0 is a scalar and v

0

is in V , for otherwise, the maximand

k(q)+

P

n

i=1

bv

i

(q) could not be expressed as the expected social utility for some probability

distribution.

10

Formally, the social surplus function is deﬁned as

W (f) = −nF +

N

X

i=0

v

i

(f). (9)

The FF would like to maximize the expected value of this social welfare with respect to

decision rule f(.), subject to nonmanipulability and weak utilitarianism. In this perspective,

we assume that they have a “prior on priors”, i.e., a distribution B on possible priors P ;

and we assume that B is uninformative — this represents the FF’s lack of knowledge about

the true distribution of citizens’ preferences. Expected social welfare can be expressed as

E

B

E

P

(W ), were W is deﬁned by (9).

4.2 The Founding Fathers’ Beliefs

The only problem is now to give formal content to the idea that the FF will choose a nonma-

nipulable f (.) so as to maximize E

B

E

P

(W ) under a vague (or non-informative) probability

B. Such a decision rule will simply be called robust. Intuitively, this can be done by a sim-

ple limiting argument, if P belongs to a family with a ﬁnite vector of parameters, by letting

the precision of B converge towards zero (or equivalently, by letting the variance-covariance

matrix of B go to inﬁnity). This deﬁnition is involved, but the intuition is simple: ﬁnd the

nonmanipulable mechanism which maximizes expected welfare under the “veil of ignorance”,

using a non-informative prior.

Auriol and Gary-Bobo (2002), (2007) have studied the existence of robust mechanisms

in this sense, assuming that the set of public decisions is ﬁnite, that individual preferences

proﬁles can be any vector and that these vectors are multivariate normal (i.e., P is multivari-

ate normal, according to the Founding Fathers’ beliefs). Thus, the domain of preferences is

general, but a normality assumption is used. As in portfolio theory, we can weaken the nor-

10

In other words, bv

0

(q)+

P

n

i=1

bv

i

(q) is proportional to a weighted average of utilities for all b and v

0

.

15

mality requirement, but will obtain a tractable model only if utility is assumed quadratic.

We follow this direction here, because our theory can easily be illustrated in the classic

quadratic-preference setting.

Assumption 5. (Quadratic preferences) Decision q is a real number, and,

V =

v(q) = θq −

q

2

2

, θ ∈ R

. (10)

In this simple setting, the true probability distribution P is just a one-dimensional distribu-

tion of the taste parameter θ, with a ﬁnite mean µ

P

, and a ﬁnite variance σ

2

P

. In this case,

we also assume that the FF do not know (µ

P

, σ

2

P

), but that they are endowed with a prior

B on possible pairs (µ

P

, σ

2

P

). In addition we assume the following:

E

B

(µ

P

) = bµ, E

B

(σ

2

P

) = bσ

2

, and V ar

B

(µ

P

) = bz

2

, (11)

where bµ, bσ

2

, bz

2

are themselves ﬁnite, and where bµ is the mean of the possible means, bσ

2

is

the mean of the possible variances, and bz

2

is the variance of the possible means. The prior

variance of θ, from the FF’s point of view, is denoted V ar

F F

(θ), and admits the well-known

decomposition,

V ar

F F

(θ) = V ar

B

[E(θ|P )] + E

B

[V ar(θ|P )]

= bz

2

+ bσ

2

.

We propose the following simple formal deﬁnition.

Deﬁnition 5 (Robust Representation Mechanism). A mechanism (f, t) is robust if it is the

limit of a sequence (f

k

, t

k

) of mechanisms, such that each (f

k

, t

k

) maximizes E

B

k

(E

P

W ) on

the set of nonmanipulable mechanisms, where (B

k

) is a sequence of priors with the property

that that bz

2

k

goes to +∞, while bσ

2

k

/bz

2

k

goes to zero.

To understand this deﬁnition, assume that all possible P distributions have the same variance

σ

2

P

= bσ

2

, but that their mean µ

P

is unknown to the FF. To approach complete ignorance,

we let the variance of the possible means, i.e. bz

2

, go to inﬁnity. As indicated above, a more

general deﬁnition is of course possible, but would be more technical.

16

4.3 Derivation of the Robust Mechanism in the Case of Quadratic

Utility

Under Assumption 5, nonmanipulability and weak utilitarianism force us to choose a utility

function of the form v

0

(q) = αq − q

2

/2 with a weight β ≥ 0 and a decision rule f

∗∗

(.), such

that

f

∗∗

(

b

θ

1

, ...,

b

θ

n

) ∈ arg max

q

(

q

n

X

i=1

b

θ

i

−

nq

2

2

+ β

αq −

q

2

2

)

, (12)

assuming that each representative i reports

b

θ

i

. We immediately ﬁnd

f

∗∗

(

b

θ

1

, ...,

b

θ

n

) =

P

n

i=1

b

θ

i

+ αβ

n + β

. (13)

Let now W

P

(α, β) be the expected welfare for a given distribution P and f

∗∗

as above. We

have

W

P

(α, β) = E

P

(

f

∗∗

(

b

θ)

N

X

i=0

θ

i

−

(N + 1)f

∗∗

(

b

θ)

2

2

)

− nF. (14)

We then compute the expected value of W

P

with respect to the FF’s prior B. Some compu-

tations yield the following formula.

Lemma 1.

E

B

[W

P

(α, β)] =

n + β −

N + 1

2

nbσ

2

(n + β)

2

+

b

2

(N + 1)

2(n + β)

2

(2αbµ − α

2

)

+

n(N + 1)

(n + β)

2

n

2

+ β

(bµ

2

+ bz

2

) − nF. (15)

For proof, see the appendix.

For given B, the best mechanism is obtained as a maximum of W = E

B

[W

P

(α, β)] with

respect to (α, β). We ﬁnd the following result.

Lemma 2. For given B, the optimal values of α and β are α

∗∗

= bµ, and

β

∗∗

=

(N + 1 − n)bσ

2

bσ

2

+ (N + 1)bz

2

. (16)

For proof, see the appendix.

17

This solution can be rewritten as a function of the ratio ζ = bσ

2

/bz

2

. We immediately ﬁnd

the limit of β

∗∗

as ζ → 0,

lim

ζ→0

β

∗∗

= lim

ζ→0

(N + 1 − n)ζ

ζ + (N + 1)

= 0.

Under decentralized knowledge, the only robust mechanism entails v

0

(q) = bµq − q

2

/2 and

β

∗∗

= 0 and therefore, the arbitrary function k must be set identically equal to 0. This

mechanism is a sampling Groves mechanism. To make a public decision, it relies on the

representatives’ reports only. Formally, we have just proved the following result.

Proposition 2. Under Assumptions 1-5, the only robust mechanism f

∗∗

(bv) maximizes

P

n

i=1

bv

i

(q), with transfers t given by (8) above.

Since preferences are assumed quadratic, we get q

∗∗

= f

∗∗

(bv

1

, ..., bv

n

) = (1/n)

P

n

i=1

b

θ

i

. In

fact, the same sampling Groves mechanism is robust in our sense with a much more general

set of preferences, but at the cost of some normality assumption (on P , not on B).

11

The sampling Groves mechanism solves a number of diﬃcult problems of a representa-

tive democracy simultaneously. It saves on the costs of producing information on preferences,

captured by the ﬁxed cost F , because of sampling; it ensures honest revelation of their pref-

erences by representatives in a very strong sense (i.e., Groves mechanisms are revealing in

dominant strategies); and ﬁnally, once subjected to the incentive transfer system (8) (see

also Proposition A1 in the appendix), every representative adheres to the same social ob-

jective (i.e., every representative agrees with the objective of maximizing

P

n

i=1

bv

i

(q)). The

interpretation of this result is that the legislative bargaining process yields an approximate

Pareto optimum, insofar as the representation is a correct mirror image of the population’s

preferences. Of course, this nice solution is obtained for a somewhat simpliﬁed economy with

quasi-linear preferences (i.e., a public good economy with possibilities of compensation).

Remark that, if we let the prior’s variance bz

2

go to zero instead, while bσ

2

remains

bounded, then, we ﬁnd lim

ζ→∞

β

∗∗

= N + 1 − n . This means that the FF know the

distribution of preferences in society for sure. In this case, the recommended solution is the

11

Again, see Auriol and Gary-Bobo (2007).

18

standard Bayesian mechanism of sub-section 2.2, where v

0

(q) = bµq − q

2

/2 = E

P

(v(q)) and

N + 1 − n is the appropriate weight of v

0

in the expected welfare function E[W | q, bv

1

, ..., bv

n

]

(and N + 1 − n is also the number of passive citizens). In this latter case, the sampled

agents represent only themselves, while in the robust mechanism, sampled agents are truly

representatives: they stand for the entire society. This is a major diﬀerence. We now show

that in this setting, an optimal number n

∗∗

can be interior, i.e., 0 < n

∗∗

< N + 1, in sharp

contrast with the standard Bayesian ﬁrst-best analysis presented in sub-section 2.2.

4.4 Optimal Number of Representatives

We can now compute the optimal number of representatives, denoted n

∗∗

. Substituting the

robust decision rule f

∗∗

(θ) = (1/n)

P

n

i=1

θ

i

in the expression for expected welfare yields

W =

N + 1

2

(bµ

2

+ bz

2

) +

bσ

2

2

−

1

n

−

1

N + 1

(N + 1)bσ

2

2

− nF. (17)

Deﬁne q

N+1

=

1

N+1

P

N

i=0

θ

i

. If we compute the ﬁrst-best surplus in an economy with N + 1

agents, using complete knowledge of the preference proﬁle and then take expectations, we

ﬁnd

E

B

E

P

"

q

N+1

N

X

i=0

θ

i

− (N + 1)

q

2

N+1

2

#

− nF = (N + 1)E

B

E

P

q

2

N+1

2

− nF

=

bσ

2

2

+

N + 1

2

(bµ

2

+ bz

2

) − nF. (18)

Let q

n

=

1

n

P

n

i=1

θ

i

. Under the robust mechanism, we get the following expression of welfare,

W = (N + 1)E

B

E

P

q

n

q

N+1

−

q

2

n

2

− nF. (19)

Taking the diﬀerence of expressions (18) and (19), we ﬁnd the welfare loss (with respect to

the complete information ﬁrst-best),

L(n) =

(N + 1)

2

E

B

E

P

(q

N+1

− q

n

)

2

. (20)

It is then easy to check that

L(n) =

1

n

−

1

N + 1

(N + 1)bσ

2

2

; (21)

19

and it follows that expression (17) is ﬁrst-best surplus, minus the cost of representatives,

minus the welfare loss due to the fact that some information on preferences is not reported.

The optimal number of representatives n

∗∗

trades oﬀ the cost of an additional representative

with the beneﬁt of reducing the welfare loss, i.e., n

∗∗

minimizes nF + L(n). The repre-

sentatives protect citizens against arbitrary public decisions, but there is a social cost of

representation.

Observe that the social cost of representation nF + L(n) does not depend on bz

2

(which can thus be arbitrarily large). It follows that if the FF had prior information on the

variance of preferences bσ

2

, they could compute the optimal number of representatives under

the robust mechanism. At the time of the writing of the constitution, the FF may have had

some knowledge of F , N and bσ, but have been well aware that these parameters vary with

time. The constitution should therefore allow for changes in the optimal n. In other words,

the number of seats in parliament should not be ﬁxed by the constitution.

12

The ﬁrst-order condition for a maximum of W with respect to n, viewed as a real

number, is easy to compute and yields −F + (1 + (N + 1)/2n

2

))bσ

2

= 0. From this we derive

the following result.

Proposition 3. With quadratic preferences, the optimal number of representatives is 1 plus

the integer part of

n = bσ

r

N + 1

2F

. (22)

If n is smaller than 1, we choose n

∗∗

= 1. This appears when F is very large, or bσ very

small. In this case, a single person (a “technocrat”) will make the public decision.

13

If, on

the contrary, F is small, or bσ is very large, we get n

∗∗

= N (everybody is a representative,

12

This does not mean that it should be free. In our stylized model, the rule to change the number of seats

could be ﬁxed by the constitution, while the number itself is not. In practice, it is usually possible to change

the number of representatives without amending the constitution. For instance, in France the number of

representatives is determined by an ”organic act” which is stronger than ordinary law but weaker than the

constitution.

13

But the technocrat is not a dictator, because, when bσ is small, preferences tend to be very close, and

there is a consensus about the optimal decision.

20

except the executive), and we obtain a direct democracy. In this latter case, the ﬁrst-best is

almost implemented.

14

Proposition 3’s formula suggests an econometric model of the form:

log(n) = log(bσ) + (1/2) log(N + 1) − (1/2) log(2F ) + , (23)

where is a zero-mean, random error term. This formulation is simple and natural. The

three factors determining the number of representatives are: the variance of preferences,

the size of the population, and the costs of representation. This simple model ﬁts the data

remarkably well, as we now show.

5 Empirical Assessment, on Political Data

To empirically predict the size of representative political institutions, we have assembled a

data set for a sample of 111 countries that possess a parliament or representative assemblies.

The total number of representatives, n, is expressed in numbers of individuals. It includes

all representatives at the national (or federal) level, e.g., the sum of the members of the

lower and upper houses, when a country has a bicameral legislature. We do not count

the representatives in local governments, in the member states of a federation, or in the

district or city-councils. Our point of view has been to study the determinants of national

representation sizes. The population size, denoted N in the following, is expressed in millions

of citizens. These two pieces of information were extracted from The Europa World Year

Book (1995). To ﬁx ideas, the USA are in the sample with n = 535 and N = 260.341.

France has 898 representatives (d´eput´es plus s´enateurs).

15

We have estimated the same

model separately with data relative to the 50 parliaments of the US States.

5.1 The Square-Root Model with World Data

To get a preliminary view of the empirical relevance of the theory, we have ﬁrst regressed

the total number of representatives n (expressed in numbers of individuals) on population

14

In the ﬁrst best, strictly speaking, we have n

∗

= N + 1 (see sub-section 2.2).

15

In the case of the UK, adding some 1221 peers to 651 MPs (in 1995) would have created an outlier: so

we decided not to add the Peers.

21

size N (expressed in millions of citizens). A ﬁrst regression of the form n = a + bN yields

signiﬁcant estimates of a and b, but with a poor R

2

(the adjusted R

2

is .27). By contrast,

a much better adjustment is obtained when, as suggested by theory, log(n) is regressed on

log(N) plus a constant (without any constraint). We ﬁnd the following result,

log(n) = 4.324

(75.26)

+ 0.41

(17.63)

log(N) (24)

In the above regression, t-statistics are between brackets. The adjusted R

2

is .74, and the

global F-statistic is a highly signiﬁcant 311.23. Moreover, the estimated constant, 4.324, and

the estimated coeﬃcient, 0.41, are both relatively close to the theoretical predictions which

are 6.561 = (1/2)(log(10

6

)/2 − log(2)), and 0.5 respectively. In particular, the estimated

power of N is below 1/2, but not much. The estimated constant captures some of the eﬀect

of the omitted variables. But the result is surprisingly good for such a crude regression. See

Fig.1, for a plot of n against N in the studied sample.

Insert Figure 1 here.

According to the theory, a more heterogeneous population should lead to larger par-

liaments, and countries where the cost of representation is high should have smaller ones.

It is diﬃcult to capture population heterogeneity σ

2

and the per capita (opportunity) cost

of representation F in the regression. We can only hope to ﬁnd proxies for σ and F . We

were not able to ﬁnd a database, or even international comparison studies on the social cost

to maintain a representative assembly. We have checked some national accounts in order

to get a sense of the costs involved. They are quite large. For instance, in the US, the

legislative branch funding rose from USD 2.8 billions in 2001 to USD 4.3 billions, requested

in 2007 (a 57% growth). The average annual cost of maintaining one representative can

hence be estimated in 2006 around USD 8 millions, or 210 times the US GNP per capita.

In Australia, the cost of maintaining the elected representatives in federal parliament was

estimated at AD 400 millions in 2004. This puts the average annual cost of maintaining

one representative around AD 2 millions (i.e., USD 2.6 millions), more than 100 times the

Australian GNP per capita. In Canada, the total cost was CD 468 millions in 2004-2005.

22

The average annual cost per representative is then CD 5.5 millions (i.e., USD 4.95 millions),

more than 200 times the Canadian GNP per capita. None of these amounts include the

costs of running an election (i.e., campaigning and administrative costs). It is obvious that

there is some variance in the unit cost of representation: in GDP per capita terms, US and

Canadian representatives cost twice as much as Australian representatives.

16

According to

the theory, this should play a role in determining the number of representatives, given that F

is in fact the sum of opportunity and direct costs per representative. To capture the impact

of these costs on the size of the legislative bodies, we rely on several proxies. We add the

logarithm of the GDP per capita to the regression. We also add the logarithm of the total

national tax revenue, expressed as a percentage of GDP (denoted T AXREV ). The idea

is that wealthier countries and wealthier governments will not ﬁnd it diﬃcult to maintain

a large assembly. The expected sign of the the tax-revenue variable is thus positive. The

expected sign of the per capita GNP is ambiguous: if it acts as a proxy for the opportunity

costs of representation, the coeﬃcient might as well be negative. We also add the logarithm

of the average government wages (denoted GOV W AGE), expressed as a percentage of GDP.

This variable provides an indication on the representatives’ wages, and that of their staﬀs.

It is related to the per capita cost of maintaining the assembly. The expected sign of this

wage variable is thus negative. Unfortunately, we have the wage data for 62 countries only.

We ﬁrst add the 3 variables sequentially to the log(n) regression. The results are presented

in Table 1 below.

Column (1) is just the crude regression presented above. The quality of ﬁt increases sub-

stantially with additional controls, as indicated by the adjusted R

2

of columns (2)-(5), which

is around 83%. The coeﬃcients of the GNP per capita and of the tax revenue are positive;

the coeﬃcient of the government wages is negative, as predicted by the theory. We next run

a regression with the 3 variables simultaneously, reported in column (2) of Table 1. There

are only 62 countries because of the missing wage data. We next run a regression without

the GNP variable because it is not signiﬁcant in the regression above and one without the

16

This is presumably due to the fact that, contrary to their US and Canadian counterparts, Australian

representatives do not set their own wages and beneﬁts (they are ﬁxed by an independent court).

23

wage variable, because it reduces our sample by half. Coeﬃcients are fairly robust, and

particularly the coeﬃcient on log(N): its values are closer to the theoretical prediction in

columns (2) and (5), with a value of .44, possibly because the simplest log-linear model has

an omitted-variable bias.

TABLE 1. Dependent Variable: log

n

(1) (2) (3) (4) (5)

Constant 4.32 2.98 5.46 4.28 3.98

(75.26)*** (7.9)*** (27.16)*** (10.0)*** (6.55)***

log(N) 0.41 0.44 0.4 0.41 0.44

(17.63)*** (14.71)*** (16.12)*** (16.88)*** (12.9)***

log(GNP) 0.04 0.04 0.02

(1.21) (1.53) (0.55)

log(TAXREV) 0.34 0.17 0.24

(2.98)*** (1.96)* (1.76)*

log(GOVWAGE) -0.12 -0.18

(-2.08)** (-1.82)*

DENSITY -0.0001 -0.0001 - 8.5×10

−5

(-2.48)** (-2.96)*** (2.26)**

GINI -2.53 -1.79 -1.1

(-5.48)*** (-3.79)*** (-1.73)*

ELF -1.74 -1.25 -1.37

(-3.85)*** (-3.07)*** (-2.41)**

GINI×ELF 3.54 2.69 2.95

(3.49)*** (3.06)*** (2.43)**

No. Obs. 111 62 93 93 55

R

2

0.74 0.83 0.83 0.85 0.86

Adjusted R

2

0.74 0.82 0.82 0.84 0.83

Sum squared Resid 14.5 4.25 8.89 7.74 3.26

Columns (1)–(5) were estimated by ordinary least squares. White heteroskedastic-consistent standard errors

are used to calculate t-statistics, which are reported in parentheses. Signiﬁcance is denoted by *** (1%); **

(5%); * (10%).

In order to measure population heterogeneity σ, we have tried diﬀerent variables. We

ﬁrst added population density (inhabitants per square kilometer, divided by 10,000) in the

basic regression. The intuition is that people who leave far apart do not interact much,

and may diﬀer more. They have diﬀerent ways of life, diﬀerent jobs, face diﬀerent climates,

etc. High density countries (e.g., Japan) should present less heterogeneity than low density

countries (e.g., the US). According to the theory, the sign of the density coeﬃcient should

then be negative. The data, which give the number of inhabitants per square kilometer in

24

1996, are from the World Bank (World Development Indicator 1998). As predicted by the

theory, the sign of the density coeﬃcient is negative and signiﬁcant, but it is quite small in

absolute value (to ﬁx ideas, a decrease of the density variable by 1 unit in the US would

result in one additional representative only).

It also seems reasonable to assume that countries including many diﬀerent linguistic

and ethnic groups are more heterogeneous. We thus add the ethno-linguistic fractionalization

index, denoted ELF , as an explanatory variable.

17

This index varies between 0 and 1 and

gives the probability that two randomly chosen individuals do not speak the same language.

Higher ethno-linguistic fractionalization indices may signal more heterogeneous populations

and the sign of the ELF coeﬃcient should then be positive. This index is known for 93 of

the 111 countries considered.

Another variable that might reﬂect population heterogeneity is the Gini coeﬃcient

of each country. Gini coeﬃcients provide a measure of heterogeneity in the sense that a

large coeﬃcient signals an unequal distribution of income in the population. If we admit

that more unequal societies are more heterogeneous, everything else being equal, they should

have larger representatives assemblies. The coeﬃcient of the Gini variable should then be

positive. For the sake of comparison with the ELF coeﬃcient, the data, which are from

the World Bank (World Development Indicators, 1998), are divided by 100 to be normalized

between 0 and 1. When added separately to the log(n) equation, the ELF and Gini index

coeﬃcients are both signiﬁcant and negative, which is the “wrong” sign, as shown by columns

(3)-(5) in Table 1. This seemingly contradicts the theoretical result that more population

heterogeneity in the population should be associated with a larger representation. However,

when we consider the joint eﬀect of wealth inequality and ethno-linguistic fragmentation

(i.e., the product ELF × Gini), the coeﬃcient of this interaction variable is both signiﬁcant

and positive, which is the expected sign. The net eﬀects of Gini and ELF are negative, but

the eﬀect of income inequality dampens as the level of ethno-linguistic fragmentation rises.

Symmetrically, the eﬀect of higher ethno-linguistic fragmentation is mitigated as the level of

income inequality rises.

17

Source: Easterly and Levine, World Bank, 2003

25

It is not clear that the variables used as regressors are really good proxies for the

degree of preference heterogeneity in a given country. It might well be that Gini and ELF

poorly capture the relevant aspects of heterogeneity. Clearly, other variables should be

included in the regression to better capture the eﬀect of preference heterogeneity on the size

of representations. From the statistical point of view, Gini and ELF are also potentially

endogenous variables, at least in the long run, but an instrumentation of these variables is

outside the scope of the present paper. Another possibility is of course that countries with

more unequal income distributions (and higher ELF indices) are characterized by a form of

power capture by the richest (and (or) by some ethnic groups). Some of the political regimes

considered in the sample are far from being ideal democracies.

Finally, we have added several other variables, capturing some aspect of countries’

history and legal institutions, as robustness checks.

18

The only additional variables that

turned out to be signiﬁcant are dummies, denoted DEM46 and OCDE indicating when the

country experienced 46 years of continuous democracy and a member of OCDE, respectively

(see Table A1, in the Appendix). DEM46 has a negative coeﬃcient: everything else equal,

old democracies have less representatives than young democracies. This suggests that the

number of seats in legislatures is characterized by institutional rigidity. OCDE has a positive

coeﬃcient and seems to capture the same eﬀects as T AXREV and GNP .

To sum up, results indicate that the number of representatives n is not determined

by a constant sampling rate: it increases less than proportionately with the size N of the

population, since according to our estimates on the 111 countries, n ≈ (0.3)N

0.4

(when N

is the count of inhabitants). This formula yields 475 representatives for a 100 millions of

people. This ﬁnding has been shown to be robust, and supports fairly well (it is indeed a

close variant of) the square-root theory of the optimal number of representatives derived

above.

Apart from assessing whether the normative principles underlying the analysis are at

18

P ERCEN T P ROT is the percentage of protestants in the country; OECD indicates an OECD country,

TRANS a former socialist (i.e;, transition) country; F EDERAL a country with a federal structure (e.g.,

the US, Germany); AF RICA indicates a country form the African continent; F ORM BRIT COL indicates

a former British colony and the UK; COMLAW indicates a country with a Common Law system; DEM 46

equals 1 if the country has been democratic in all 46 years (1950-1995) (see Treisman (2000)).

26

work in actual political institutions, our empirical exercise can also be viewed as a ﬁrst step

to compare the political systems of diﬀerent countries, in terms of representation size. Table

A2, in the appendix, gives the list of countries, the observed value of n and its ﬁtted value bn,

computed with the help of Regression 4 in Table 1. For the missing data the simple regression

1 in Table 1 is used. These ﬁgures are reported in italic. Figure 2 plots the actual number

n of representatives (denoted REP RE) against the predicted number of representatives bn

(denoted REP REF ). This plot has been drawn with the results of Regression 4 in Table 1.

Insert Fig. 2 here

We ﬁnd that France and Italy have “too many” representatives, whereas the United

States do not have “enough” of them (i.e., they lie below the regression line). In fact,

both France and Italy have more representatives than the US. According to our results,

the US Lower and Upper House should have 807 members instead of 535. The political

US institutions seem characterized by historical rigidity. We conﬁrm this result in the next

sub-section.

5.2 Number of Representatives in the US State Legislatures

Using the data provided by McCormick and Turner (2001), for US state legislatures in 1996,

we have tested the square-root model with the 50 US States, adding the state senators and

representatives together to form the n variable. The state population (in million) is for 2005.

We ﬁnd that the crude log-linear regression yields

log(n) = 4.696

(52.35)

+ 0.172

(3.32)

log(N), (25)

the adjusted R

2

is equal to .21, and the global F = 14.16, with exactly 49 observations

(t-statistics are in parentheses). Adjustment quality is mediocre as compared to the results

obtained with world data. Among the US states, New Hampshire, Nebraska and Nevada are

outliers. New Hampshire has a plethoric n = 400 representatives and we have removed this

state from the sample. If we take the 50 States, the simple log-linear regression above yields

a coeﬃcient of 0.14 on log(N), with a t of 2.55 , and R

2

= .12. Fig. 3, which includes New

27

Hampshire, shows that the quality of the adjustment is bad. Removing another outlier will

not change results much (this will not increase the coeﬃcient on log(N) substantially).

Insert Fig.3 here

We then added the population density in 2000 and representatives’ salary, averaged for 1995-

97 (known for 40 states from McCormick and Turner 2001). Without New Hampshire, this

yields the following regression, which is a little closer to our square-root model:

log(n) = 4.723

(41.13)

+ 0.218

(3.09)

log(N) + 5.11

(2.47)

(10

−4

)Density − 0.218

(−2.03)

(10

−6

)Salary. (26)

The adjusted R

2

= .35, the global F = 4.78 (signiﬁcant at 3%) and t-statistics are in

parentheses. We already observed that the US are an outlier among nations in the world. A

strong dependence on history, and a very slow historic speed of adjustment of the number

of representatives seem to be the main explanations for the low quality of the adjustment.

Population has increased enormously in some US states during the 20th century, without

much change in the number of state representatives. The number of federal representatives in

Washington D.C. has itself been ﬁxed by statute in 1929 (see O’Connor and Sabato (1993)).

The US seem to be characterized by an extreme form of rigidity in these matters.

These results and those obtained with world data suggest that some countries have

an excessive number of representatives while others have too few. It seems important to

analyze the impact of having too few or too many representatives on the performance of

institutions. With too few representatives, public decisions could well be biased in favor of

active minorities, to the detriment of under-represented (or less organized) groups. Casual

observations also suggest that the corruption level could be higher in countries characterized

by an “excessive” number of representatives. We indeed ﬁnd a result of that sort below.

5.3 The Number of Representatives and Red Tape

We now examine the link between the number of representatives and barriers to business

entry, entrepreneurship and trade. The Public Choice school oﬀers a theory relating lobby

activity and the number of representatives (see Mueller (2003)): the inﬂuence of each rep-

resentative should decrease with their number; lobbies would be ready to pay more to buy

28

a vote when the number of seats in parliament is low. Becker’s (1983) approach is also

compatible with the existence of an impact of the number of representatives, if n aﬀects the

pressure groups’ inﬂuence functions through changes in the “pressure technology”.

To check for the presence of a possible inﬂuence of the number of representatives on

variables related to lobbying activity, we consider in turn 3 indices: (i) a measure of trade

openness, denoted F REEOP (and taken from Barro and Lee (1994)); (ii) a measure of

the direct cost of meeting government requirements to open a new business, expressed as

a fraction of 1999 GDP per capita, denoted SUNKCOST (due to Djankov et al. (2002));

(iii) and a measure of whether state interference hinders business development, denoted

ST AT EINT ERF (due to Treisman (2000)). We regress the three variables on log(n),

log(N), log(GNP ), and many controls. The only signiﬁcant variables are LAND, which

measures the country’s land surface in millions of square kilometers, DEM46 and T RANS

dummies (deﬁned above). The regressions, presented in Tables 2 and 3, check for correlation,

not necessarily for causality. Yet, as explained above, the number of representatives is largely

predetermined and rigid in most countries: it is either ﬁxed by the constitution, or by statutes

with a high rank in the hierarchy of norms, that cannot be changed easily. It is of course

endogeneous in the long run. Hence, the number of representatives has good chances of being

”exogenous” compared to F REEOP , SUNKCOST and ST AT EINT ERF , that can be

changed easily. Table 2 and 3 presents the results obtained with OLS.

We ask the following question: is it true that the variables under study are in fact

inﬂuenced, not by log(n) itself, but by the excess number of representatives, deﬁned as the

residual of the crude log-linear regression, log(n) − log(bn) = log(n) − (0.4) log(N) − 4.32.

This hypothesis can be tested by means of the standard F -test of a single linear restiction,

because using log(n) − log(bn) as a control instead of log(n) and log(N) is tantamount to

assuming that the coeﬃcient on log(N) is equal to −0.4 times the coeﬃcient on log(n).

According to some theories, we should observe more restrictions to trade, and thus

less openness, in countries with a smaller legislature: protectionist policies would be easier

to implement with a relatively smaller parliament.

19

But the impact of log(n) on trade

19

On the other hand, since there exists a positive correlation between an economy’s exposure to interna-

29

openness is not signiﬁcant. Table 2 presents a study of the possible impact of log(n) or

(log(n) − log(bn)) on F REEOP . Column (1) in Table 2 is the unconstrained version of the

regression, while the constrained version is given by column (3). However log(n) − log(bn) is

not signiﬁcant in regression (3). Column (2) yields good results, but the signiﬁcant coeﬃcient

on log(n) is likely to indirectly capture the eﬀect of log(N), which has been removed from

the regression. Total population and land surface are signiﬁcant with the expected negative

sign, as shown by column (4). This result is robust to the addition of many controls.

20

We

ﬁnd more interesting results with the other indices.

TABLE 2

Dependent Variable: FREEOP

(1) (2) (3) (4)

Constant 0.09 0.22 0.11 0.14

(1.3) (3.8)*** (3.33)*** (5.39)***

log(n) 0.01 -0.024

(0.83) (-2.39)**

log(N) -0.02 -0.02

(-3.09)*** (-3.09)***

log(GNP) 0.015 0.017 0.02 0.02

(3.76)*** (4.65)*** (3.05)*** 4.48***

LAND -1.6×10

−5

-1.89×10

−5

-2.19×10

−5

-1.65×10

−5

(-5.3)*** (-5.31)*** (-5.5)*** (-5.37)***

DEM46 0.05 0.054 0.048 0.05

(2.51)** (2.53)*** (2.53)** (2.69)***

log(n)-log(bn) 0.01

(0.49)

No. Obs. 67 67 67 67

R

2

0.65 0.61 0.57 0.64

Adjusted R

2

0.62 0.58 0.54 0.62

Sum squared Resid 0.13 0.15 0.16 0.13

All columns were estimated by ordinary least squares. White heteroskedastic-consistent standard errors are

used to calculate t-statistics, which are reported in parentheses. Signiﬁcance is denoted by *** (1%); **

(5%); * (10%).

The ﬁrst dependent variable studied in Table 3 , ST AT EINT ERF , provides a mea-

tional trade and the size of the government (e.g. Rodrik (1998), Alesina and Wacziarg (1998)), FREEOP

could presumably be positively related to the number of representatives.

20

The result holds when we sequentially add: AFRICA, COMMONLAW, ELF, FEDERAL, FORMBRIT-

COL, PERCENTPROT, GOVWAGE. None of these variables have a coeﬃcient signiﬁcantly diﬀerent from

zero.

30

sure of barriers to business for existing ﬁrms (i.e., whether State interference hinders the

development of business). According to some theories, we should observe larger barriers

to business in countries with a smaller legislature. The rent-seeking strategies of lobbies

would be easier to carry out with a smaller number of seats in parliament. Yet, in Table

3, column (1a) exhibits a positive and signiﬁcant coeﬃcient on log(n). This result is robust

to the addition of many controls.

21

It seems that the residual of the regression of log(n) on

log(N) is in fact the appropriate explanatory variable. Column (1a) is the unconstrained

version of the regression, and column (1b) is the constrained version, exhibiting a positive

and signiﬁcant coeﬃcient on (log(n) − log(bn)). The F -test testing the model of column (2b)

against that of column (2a) yields (45 − 5)(13.32 − 12.32)/12.32 = 3.246. The critical value

being F

95

(1, 40) = 4.08 at the 5% level, we cannot reject the assumption that it is the excess

number of representatives, log(n) − log(bn), which has an impact on ST AT EINT ERF .

The second variable, SUNKCOST , measures barriers to entry for entrepreneurs

(i.e., barriers to the creation of new ﬁrms). This variable has been shown to be a major

determinant of the size of a country’s informal sector, and also to contribute to the level of

rents in the legal sector (see Auriol and Warlters (2005), Ciccone and Papaioannou (2007)).

Again, we have reasons to expect higher barriers to entry in countries with relatively smaller

legislatures. Yet, column (2a) in Table 3 shows the opposite result: the coeﬃcient on log(n)

is positive and signiﬁcant at the 5% level. The result is again robust to the addition of many

controls.

22

It is also robust when the regression is run without France and Italy, which have

been identiﬁed as outliers above.

23

The unconstrained regression is given by column (2a),

while the constrained regression, with log(n) − log(bn) as a regressor, is given by column

(2b). Note that log(n) − log(bn) has a signiﬁcant coeﬃcient in column (2b). The F -test

comparing columns (2a) and (2b) is (71 − 5)(20.46 − 20.05)/20.05 = 1.35, and the critical

21

The result holds when we sequentially add, AFRICA, ELF, COMMONLAW, LAND, FEDERAL,

FORMBRITCOL, PERCENTPROT, GINI, GOVWAGE, OECD, TRANS, to the regression.None of theses

variables have a signiﬁcant coeﬃcient.

22

The result holds when we sequentially add to the regression:AFRICA, COMMONLAW, DEM46, LAND,

FEDERAL, FORMBRITCOL, GINI, GOVWAGE. None of these variables have a coeﬃcient which is signif-

icantly diﬀerent from zero.

23

The coeﬃcient of the log-number of representatives is then positive and signiﬁcant at the 1% level.

31

value is F

95

(1, 66) ≈ 4, so we cannot reject the assumption that it is the residual of the crude

log-linear regression which has an impact on SUNKCOST .

TABLE 3

Dependent Variable: STATEINTERF STATEINTERF SUNKCOST SUNKCOST

(1a) (1b) (2a) (2b)

Constant 1.58 4.65 0.56 2.32

(1.45) (6.22)*** (1.18) (3.38)***

log(n) 0.47 0.45

(2.2)** (2.26)**

log(N) -0.08 -0.25

(-0.63) (-2.29)**

log(GNP) -0.19 -0.26 -0.23 -0.23

(-2.31)** (-3.0)*** (-3.05)*** (-3.07)***

DEM46 -0.4 -0.38

(-2.27)** (-1.81)*

TRANS -0.34 -0.3

(-2.16)** (-2.07)**

log(n)-log(bn) 0.51 0.42

(2.08)** (2.22)**

No. Obs. 45 45 71 71

R

2

0.43 0.38 0.3 0.28

Adjusted R

2

0.37 0.34 0.25 0.25

Sum squared Resid 12.32 13.32 20.05 20.46

All columns were estimated by ordinary least squares. White heteroskedastic-consistent standard errors are

used to calculate t-statistics, which are reported in parentheses. Signiﬁcance is denoted by *** (1%); **

(5%); * (10%).

The results of Table 3 suggest that it is the part of log(n) which is unexplained by

total population (i.e., the excess number of representatives) which is in fact causing a higher

level of barriers to entry. These results show a negative correlation between the number of

representatives and the degree of laissez-faire (or free-market orientation) of a country. We

are not aware of a theory explaining these facts, but they suggest a possible straightforward

“quantity theory” of the legislators’ activity and meddling in the operation of markets: the

more representatives, the more people work on law and regulation; the higher their “output”,

and the more they meddle in business activity. A related idea is provided by Becker (1983,

p 388):

“Cooperation among pressure groups is necessary to prevent the wasteful ex-

32

penditures on political pressure that result from the competition for inﬂuence.

Various laws and political rules may well be the result of cooperation to reduce

political expenditures, including restrictions on campaign contributions and the

outside earnings of Congressmen, the regulation of and monitoring of lobbying

organizations, and legislative and executive rules of thumb that anticipate (and

thereby reduce) the production of pressure by various groups.”

To this list, we add the number of representatives itself. It might well be that in some

countries, the low number of representatives is a long-established, endogenous response of

the political system, reﬂecting cooperation among various forces to reduce lobbying and

ineﬃcient state interventions. If these ideas are true, there are some chances that we will

ﬁnd a positive correlation between the number of representatives and corruption.

5.4 The Number of Representatives, and Corruption

We ﬁnally study the possible impact of the number of representatives on the level of perceived

corruption in a country. Treisman (2000) has analyzed the causes of perceived corruption in a

cross-national study, using T ISCORE, a well-known corruption index computed by Trans-

parency International. Treisman (2000) ﬁnds that countries with more developed economies,

protestant traditions, histories of British rule, long exposure to democracy and higher im-

ports are less “corrupt”. Federal states are more “corrupt”. We have reproduced Treisman’s

regressions with additional controls and chieﬂy, we added the number of representatives in

the regressions.

The results are presented in Table 4; they strongly suggest that the number of representatives

has a positive impact on corruption. Columns (1), (2), and (3) of Table 4 reproduce columns

(2), (3), and (4), respectively, in Treisman (2000, table 3, p 58), for the year 1996, with the

addition of the number of representatives n.

24

With these changes, contrary to Treisman’s

results, F ORMBRIT COL is not signiﬁcant, while DEM46 becomes signiﬁcant.

25

24

The dataset is not exactly the same: more countries are included in our results. Some variables are

diﬀerent: we use F REEOP instead of the imports/GNP ratio in Treisman’s work. RAW M AT is the

percentage of raw materials in the country’s exports.

25

This is true, with or without the number of representatives.

33

TABLE 4

Dependent Variable: TISCORE

(1) (2) (3) (4)

Constant 14.94 13.69 13.82 12.17

(13.28)*** (11.31)*** (10.14)*** (10.21)***

n 0.0018 0.0017 0.002 0.0023

(2.61)** (2.07)** (1.93)* (3.51)***

COMLAW -0.18 -0.17 -0.12

(-0.45) (-0.46) (-0.32)

FORMBRITCOL -0.72 -0.56 -0.64

(-1.61) (-1.28) (-1.46)

PERCENTPROT -0.026 -0.014 -0.01 -0.01

(-3.85)*** (-2.15)** (-1.77)* (-3.22)***

ELF -0.24 -0.2 -0.42

(-0.33) (-0.31) (-0.57)

RAWMAT -0.008 -0.013 -0.01

(-0.69) (-1.19) (-1.16)

log(GNP) -1.29 -1.02 -1.03 -0.81

(-9.24)*** (-6.8)*** (-5.85)*** (-5.08)***

FEDERAL 0.95 0.93

(2.35)** (2.06)**

DEM46 -1.53 -1.72 -1.63

(-3.34)*** (-3.57)*** (-4.73)***

FREEOP 0.3

(0.13)

AFRICA -1.12

(-2.81)***

OECD -0.76

(-1.95)*

DENSITY -0.0005

(-4.73)***

No. Obs. 69 69 56 79

R

2

0.70 0.75 0.76 0.77

Adjusted R

2

0.67 0.71 0.71 0.75

Sum squared Resid 115.03 96.71 80.65 101.74

Columns (1) to (4) were estimated by ordinary least squares. White heteroskedastic-

consistent standard errors are used to calculate t-statistics, which are reported in

parentheses. Signiﬁcance is denoted by *** (1%); ** (5%); * (10%).

But the novelty is of course that the coeﬃcient on n is positive and signiﬁcant in the 4

variants presented in Table 4. The result is thus fairly robust.

26

We are tempted to interpret

26

The signiﬁcant coeﬃcient on n in column (4) is robust to the inclusion of many controls, i.e., ELF,

COMLAW, LAND, FEDERAL, FORMBRITCOL, GINI, GOVWAGE, TRANS. None of these variables

have signiﬁcant coeﬃcients.

34

this fact as a conﬁrmation of the “quantity theory” sketched above. More representatives

produce more red tape and induce more corruption.

To sum up, putting together the results of Tables 3 and 4, suggests that the number

of representatives really matters. Political regimes in which the rate of representation is low,

the inﬂuence and “value” of each representative are correlatively high, could paradoxically

be regimes in which the supply of intervention is less elastic, and the occasions for corruption

more limited. These results are of course just an indication that the subject deserves more

attention. Additional work is needed to check for robustness and causality.

6 Conclusion

We have proposed a model of a representative democracy, based on a two-stage model of

constitution design, with a constitutional and a legislative stage. This model embodies

a notion of political stability of the constitution, called robustness, which emphasizes the

idea that the founding fathers do not know the distribution of citizens’ preferences. From

this model, we derived a “square-root formula” for the number of representatives, stating

that the optimal number should be proportional to the square root of total population.

Regression work on a sample of more than a 100 countries shows that the number of national

representatives is proportional to total population to the power of 0.4 : the square-root

theory is almost true. We then ﬁnd that the USA is an outlier with too few representatives,

while France and Italy, for instance, have too many. The quality of ﬁt is lower when data

on the 50 US state legislatures is used. We ﬁnally cannot reject the assumption that the

excess number of representatives has an impact on the degree of state interference and on

an index of barriers to entry of new ﬁrms (i.e., red tape). The number of representatives

itself has a signiﬁcant, positive impact on the degree of perceived corruption. The number

of representatives thus matters and we suggest that a “quantity theory” of representatives

hold: more seats in parliament are associated with more red tape, more state interference in

business, and more corruption.

35

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39

8 Appendix

The formal statement of the result used in the derivation of the theorem is as follows.

Proposition A1. Assume that there are at least three possible decisions, that the separation-

of-powers, subsidiarity and anonymity principles hold. Assume that any utility function v

is possible ( V is a universal domain). Then, (f, t) is nonmanipulable if and only if the

following three conditions hold:

f(bv) ∈ arg max

q∈Q

(

k(q) +

n

X

i=1

bv

i

(q)

)

(27)

where k is an arbitrary ﬁxed function, and for all i = 1, ..., n;

t

i

(bv) = −

X

j6=i

bv

i

(f(bv)) − k(f(bv)) + m(bv

−i

), (28)

where m is an arbitrary ﬁxed function that doesn’t depend on v

i

; and ﬁnally,

k, and m are ﬁxed in the constitution. (29)

See Auriol and Gary-Bobo (2007) for a proof of this result, which is an adaptation of the

classic characterization of dominant strategy mechanisms, under the assumption of quasi-

linear preferences. The hard part in the proof of this proposition is the “only if” part;

it heavily relies on K. Roberts’ (1979) characterization theorem. It is intuitive that the

requirement of dominant strategies restricts the set of admissible mechanisms in such a way,

even if it is diﬃcult to prove that these mechanisms are the only nonmanipulable ones. Note

that these mechanisms are also budget-balanced by construction, because there is at least

one citizen which is not a representative (i.e., at least agent 0 does not report about his

(her) preferences). We further assume that the FF are constrained to choose f(.) in the set

deﬁned by Proposition A1 above, even if the domain V is restricted to a particular class of

utility functions (to keep matters simple.

27

) We now provide a short proof of the Lemmas.

27

In the case of quadratic preferences, it is well-known that there exists a fully optimal, budget-balanced,

Groves mechanism: but it is a member of the same family (see Moulin (1988), chapter 8, Groves and Loeb

(1975)). In the quadratic case, we can design the transfers so as to “isolate” the representatives: they can

self-ﬁnance their revelation incentives.

40

Proof of Lemma 1. Let

P

N

i=0

θ

i

= r + s,where r =

P

n

i=1

θ

i

and s = θ

0

+

P

N

i=n+1

θ

i

. Then,

W

P

(a, b) = E

P

(r + s)

r + ab

n + b

−

(N + 1)(r + ab)

2

2(n + b)

2

− nF.

Using the fact that E

P

(r) = nµ

P

, E

P

(s) = (N + 1 − n)µ

P

, E

P

(rs) = E

P

(r)E

P

(s) (because

of independence), E

P

(r

2

) = nσ

2

P

+ n

2

µ

2

P

and after some elementary computations, we ﬁnd

W

P

(a, b) =

1 −

N + 1

2(n + b)

nσ

2

P

n + b

+

n(N + 1)

(n + b)

2

n

2

+ b

µ

2

P

+

ab

2

(N + 1)

(n + b)

2

µ

P

−

a

2

b

2

(N + 1)

2(n + b)

2

− nF. (30)

We then take the expectation of W

P

(a, b) with respect the the prior distribution B. This

yields the stated result.

Q.E.D.

Proof of Lemma 2. We ﬁrst maximize the expression of E

B

[W

P

(α, β)] given by Lemma 1

with respect to a. This is equivalent to maximizing (αbµ − α

2

/2). Hence, α

∗

= bµ. Substitute

next α

∗

= bµ in W (α, β) ≡ E

B

[W

P

(α, β)]. We then easily obtain

E

B

[W

P

(bµ, β)] =

n + β −

N + 1

2

nbσ

2

(n + β)

2

+

n(N + 1)

(n + β)

2

n

2

+ β

(bµ

2

+ bz

2

)

+

bµ

2

β

2

(N + 1)

2(n + β)

2

− nF. (31)

We ﬁnally maximize W (bµ, β) with respect to β. After some simpliﬁcations, we ﬁnd the

ﬁrst-order condition

∂W (bµ, β)

∂β

=

n

(n + β)

3

bσ

2

(N + 1 − n − β) − bz

2

β(N + 1)

= 0.

We then solve this equation for β

∗

. It is easy to check that W is strictly quasi-concave and

it follows that β

∗

is the unique global maximizer of W (bµ, β).

Q.E.D.

41

TABLE A1. Dependent Variable: log

n

(1) (2) (3) (4) (5)

Constant 3.4*** 4.21*** 4.03*** 4.22*** 4.02***

(7.16) (9.97) (9.34) (10.21) (11.67)

log(N) 0.45*** 0.41*** 0.4*** 0.4*** 0.39***

(15.27) (16.79) (16.66) (16.16) (18.32)

log(GNP) 0.08** 0.07** 0.06* 0.04 0.02

(2.11) (2.35) (1.69) (1.09) (0.76)

log(TAXREV) 0.3*** 0.16* 0.12 0.16* 0.15*

(3.19) (1.94) (1.37) (1.94) (1.91)

DENSITY -0.0001** -0.0001*** -0.0001*** - 0.0001*** - 0.0001***

(-2.27) (-3.44) (-2.82) (-2.67) (-2.9)

GINI -1.63*** -2.1*** -1.37** -1.57*** -0.71**

(-3.49) (-4.32) (-2.24) (-2.72) (-2.21)

ELF -1.22*** -1.31*** -0.83* -0.98**

(-2.67) (-3.15) (-1.74) (-2.07)

GINI×ELF 2.7*** 2.89*** 1.69 2.18**

(2.74) (3.26) (1.6) (2.16)

DEM46 -0.19 -0.21* -0.22* -0.24** -0.2*

(-1.27) (-1.92) (-1.82) (-2.31) (-1.96)

PERCENTPROT -0.0006

(-0.34)

FEDERAL 0.009

(0.085)

COMLAW -0.13

(-1.1)

FORMBRITCOL 0.06

(0.54)

OECD 0.24** 0.26** 0.31***

(2.08) (2.13) (3.11)

TRANS 0.10

(0.85)

AFRICA 0.14

(1.28)

No. Obs. 71 93 93 93 111

R

2

0.88 0.86 0.87 0.86 0.83

Adjusted R

2

0.86 0.84 0.85 0.85 0.82

Sum squared Resid 4.51 7.4 6.9 7.06 9.5

Columns (1)–(5) were estimated by ordinary least squares. White heteroskedastic-consistent standard errors

are used to calculate t-statistics, which are reported in parentheses. Signiﬁcance is denoted by *** (1%); **

(5%); * (10%). PERCENTPROT is the percentage of protestants in the country; OECD indicates an OECD

country, TRANS a former socialist (i.e, transition) country; FEDERAL a country with a federal structure

(e.g., the US, Germany); AFRICA indicates a country form the African continent; FORMBRITCOL indicates

a former British colony and the UK; COMLAW indicates a country with a Common Law system; DEM46

equals 1 if the country has been democratic in all 46 years (1950-1995) (source Treisman (2000)).

42

Table A2: Actual and ﬁtted number of representatives

Value bn has been computed based on regression 4 in Table 1. For missing data, reported in

italic, regression 1 in Table 1 has been used.

COUNTRY n bn

Albania 140 113.21610

Angola 220 187.31671

Argentina 329 285.41909

Armenia 190 129.68857

Australia 219 308.46221

Austria 247 293.03681

Azerbaijan 350 170.93651

Bangladesh 300 460.99839

Belgium 221 262.90633

Benin 83 121.36892

Bolivia 157 140.49486

Bosnie-Herz 240 128.93847

Brazil 594 463.96416

Bulgaria 240 213.43878

Burkina-Faso 227 155.45687

Cambodia 120 144.75544

Cameroon 180 161.83940

Canada 399 336.39994

Cl Africa Rep. 85 79.448245

Chile 167 178.12343

Colombia 267 233.24201

Costa Rica 57 114.81261

Coast Ivory 175 188.57065

Croatia 201 143.04017

Czech Rep. 281 251.82664

Denmark 175 243.65874

Dominican Rep. 150 138.41592

Egypt 664 460.33356

El Salvador 84 113.00308

Equ. Guinea 80 39.270870

Estonia 101 89.256839

Fiji 104 63.739114

Finland 200 218.41965

France 898 545.84014

Gabon 120 80.252787

Germany 740 661.84103

Ghana 200 182.14392

Greece 300 231.76985

Grenada 28 24.882613

43

COUNTRY n bn

Guatemala 116 142.01617

Guyana 65 69.559220

Honduras 128 117.29668

Hungary 386 277.03339

Iceland 63 54.259966

India 790 860.18700

Indonesia 500 542.36577

Ireland 226 165.18585

Israel 120 175.02468

Italy 945 570.24767

Jamaica 81 103.79495

Japan 763 704.17869

Jordan 120 141.44161

Kazakhstan 177 238.86700

Kenya 188 239.66050

Rep. of Korea 299 415.62166

Kyrgyr. Rep. 105 139.26255

Latvia 100 110.94319

Lebanon 128 84.179405

Lesotho 65 82.219469

Lithuania 141 129.08048

Macedonia 120 98.900478

Madagascar 138 157.81119

Malawi 177 155.33208

Malaysia 212 239.34138

Mali 129 145.56781

Mauritania 135 90.158273

Mauritus 62 67.472499

Mexico 628 389.86490

Moldava 104 137.72470

Mongolia 76 112.42436

Mozambiq. 250 165.65331

Nanibia 72 91.836596

Nepal 255 198.00898

Netherlands 225 325.95926

New Zealand 99 152.13716

Nicaragua 92 116.27401

Niger 83 127.39640

Norway 165 219.39632

Pakistan 304 448.11270

Panama 72 89.919477

Papua Guin. 109 121.11107

Paraguay 125 105.57714

44

COUNTRY n bn

Peru 120 239.17824

Phlippines 250 351.30956

P